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→The force and diffusion terms “force”项与“diff”项
Consider particles described by ''f'', each experiencing an ''external'' force '''F''' not due to other particles (see the collision term for the latter treatment).
Consider particles described by ''f'', each experiencing an ''external'' force '''F''' not due to other particles (see the collision term for the latter treatment).
Suppose at time ''t'' some number of particles all have position '''r''' within element <math> d^3\bf{r}</math> and momentum '''p''' within <math> d^3\bf{p}</math>. If a force '''F''' instantly acts on each particle, then at time ''t'' + Δ''t'' their position will be '''r''' + Δ'''r''' = '''r''' + '''p'''Δ''t''/''m'' and momentum '''p''' + Δ'''p''' = '''p''' + '''F'''Δ''t''. Then, in the absence of collisions, ''f'' must satisfy<syntaxhighlight lang="latex">
Suppose at time ''t'' some number of particles all have position '''r''' within element <math> d^3\bf{r}</math> and momentum '''p''' within <math> d^3\bf{p}</math>. If a force '''F''' instantly acts on each particle, then at time ''t'' + Δ''t'' their position will be '''r''' + Δ'''r''' = '''r''' + '''p'''Δ''t''/''m'' and momentum '''p''' + Δ'''p''' = '''p''' + '''F'''Δ''t''. Then, in the absence of collisions, ''f'' must satisfy
<math>
f\left ( \textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t \right )\, d^{3}\textbf{r}\, d^{3}\textbf{p}= f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}
f\left ( \textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t \right )\, d^{3}\textbf{r}\, d^{3}\textbf{p}= f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}
</math><syntaxhighlight lang="mathematica">
</math>
Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume <math> d^3\bf{r}</math> '<math> d^3\bf{p}</math> changes, so
Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume <math> d^3\bf{r}</math> '<math> d^3\bf{p}</math> changes, so